An introduction to Dynamic Energy Budget (DEB) …dimension unit energy per unit time) can thus be written as p˙A ={p˙ Am}fV2/3,where{p˙Am }isthemaximum surface-area-specific assimilation

56 (2006) 85–102www.elsevier.com/locate/seares

Journal of Sea Research

An introduction to Dynamic Energy Budget (DEB) models withspecial emphasis on parameter estimation

Jaap van der Meer

Royal Netherlands Institute for Sea Research (NIOZ), PO Box 59, 1790 AB Den Burg, Texel, The Netherlands

Received 20 November 2005; accepted 6 March 2006Available online 12 March 2006

Abstract

Theories of dynamic energy budgets (DEB) link physiological processes of individual organisms, such as ingestion,assimilation, respiration, growth and reproduction, in a single framework. In this introduction, I summarise the most encompassingDEB theory developed so far [Kooijman, S.A.L.M., 2000. Dynamic Energy and Mass Budgets in Biological Systems. CambridgeUniv. Press, Cambridge.] and compare it with various alternative approaches. I further review applications of the DEB model toparticular species and discuss what sort of data sets are needed and have been used to estimate the various model parameters.Finally, I argue that more comparative work, i.e. applying DEB models to a wide range of species, is needed, to see whether we canunderstand the variability in parameter values among species in terms of their ecology and phylogeny.© 2006 Elsevier B.V. All rights reserved.

Keywords: Dynamic Energy Budgets; Metabolic theory of ecology; Scope for growth; Mytilus edulis

1. Introduction

A dynamic energy budget (DEB) model of anindividual organism describes the rates at which theorganism assimilates and utilises energy for mainte-nance, growth and reproduction, as a function of thestate of the organism and of its environment (Nisbet etal., 2000; Kooijman, 2001).The state of the organismcan be characterised by, for example, age, size andamount of reserves, and the environment by e.g. fooddensity and temperature. Dynamic energy budgetmodels of individual organisms can be used as thebasic building blocks in model studies of the dynamicsof structured populations (Metz and Diekmann, 1986).Practical applications of DEB models include the

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1385-1101/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.seares.2006.03.001

optimisation of pest control (Van Oijen et al., 1995),the development of optimal harvesting strategies,including the harvesting of microbial products suchas penicillin, and the reduction of sludge production insewage treatments (Ratsak, 2001).

One of the most encompassing theories of dynamicenergy budgets is the DEB theory developed by BasKooijman in the 1980s (Kooijman, 1986a, 2000). Thistheory resulted in the so-called κ-rule DEB model,which assumes that the various energetic processes,such as assimilation rate and maintenance, aredependent either on surface area or on body volume.The model further assumes that the assimilatedproducts first enter a reserve pool, from which theyare allocated to maintenance, growth and reproduction.The κ-rule says that a fixed fraction κ is allocated tomaintenance and growth and that the remaining

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Fig. 1. Schematic representation of the κ-rule DEB model. Part of theingestion is assimilated, the rest is lost as faeces. The assimilatedproducts enter the reserve compartment. A fixed fraction κ of the fluxfrom the reserves is spent on maintenance, heating (for endotherms)and growth (with a priority for maintenance), the rest goes to maturity(for embryos and juveniles) or reproduction (for adults) and maturitymaintenance.

86 J. van der Meer / Journal of Sea Research 56 (2006) 85–102

fraction 1-κ is available for development and repro-duction. Though the κ-rule model is based on just afew straightforward assumptions concerning energyacquisition, reserve dynamics, energy allocation andmaintenance, it is generally considered to be complex.For example, Brown et al. (2004a) states that DEBmodels are complex, using many variables andfunctions. He claims that there is room for a comple-mentary and even more general approach. This apparentcomplexity and intractability may have hamperedwidespread application and testing of DEB models.

Aim of the present paper is therefore twofold. First,it intends to provide an easily accessible introductionto the basics of the κ-rule DEB model. From thisintroduction it should become clear that the DEBmodel is as general and simple as possible, withoutlosing the essentials of the energy budget of anindividual organism. An even more general approachdoes not seem feasible. Second, the paper should be ofhelp to the practically inclined biologist who aims tolink empirical observations on the physiology andenergetics of a particular species to a theoreticalenergy budget model. In the second half of the paperemphasis is therefore put on practical aspects ofestimating DEB parameters, using observational andexperimental data.

2. DEB theory

2.1. Introduction

Kooijman's DEB theory describes the individualorganism in terms of two state variables: structural body,quantified as volume V, and reserves, quantified asenergy density [E]. The latter variable gives the amountof reserves E per unit of the structural body volume V.The square brackets in the notation of energy densityindicate that the variable is expressed per unit volume.Appendix A provides more information on the notationused in DEB theory. The structural body and thereserves both have a constant chemical composition.This assumption is called the assumption of stronghomeostatis. The amount of reserves can change relativeto the amount of structural body, for example as a resultof variable food conditions. This implies that thechemical composition of the total body may change. Italso implies that a general model of the dynamic energybudget of an individual must distinguish betweenstructural body and reserves. Organisms are able tosurvive prolonged periods of starvation, during whichthey continuously have to allocate energy to mainte-nance. These energy needs cannot be immediately

fulfilled from feeding, which is another argument infavour of taking reserves explicitly into account.

DEB theory assumes that assimilates derived fromingested food directly enter the reserves. A fixedfraction κ (read kappa) of the energy utilised from thereserves is spent on growth and somatic maintenance,the rest on development and reproduction. Based on thisrule the DEB growth model has been called the κ-rulemodel (Fig. 1). Priority is always given to somaticmaintenance, and if the energy utilisation rate from thereserves is no longer sufficient to pay for the somaticmaintenance costs, the individual dies.

The energy ingestion rate JX is proportional to thesurface-area of the organism V2/3 and is related to fooddensity through a Hollings type II curve. Hence JX={JXm}fV

2/3, where {JXm} is the maximum ingestion rateper unit of surface area and f is the scaled functionalresponse. The scaled functional response, which canvary between 0 and 1, is given by f≡X/(XK+X), where Xis the food density in the environment and XK thesaturation coefficient. The saturation coefficient, orMichaelis-Menten constant, is the food density at whichthe ingestion rate is half the maximum. In terms ofHollings type II functional response it is equal to thereciprocal of the product of the area of discovery (orsearching rate) and the handling time (Fig. 2). DEBtheory assumes that the assimilation efficiency of food isindependent of feeding rate. The assimilation rate (withdimension unit energy per unit time) can thus be writtenas p ˙A={p ˙Am}fV

2/3, where {p ˙Am} is the maximumsurface-area-specific assimilation rate. The precise

Fig. 2. The ingestion rate as a function of food density is described as aHolling's type II functional response, with the underlying idea that theanimal is either searching for prey or handling prey. Searching occursat a constant rate, and each prey requires a constant (expected)handling time. The initial slope of the function is given by thesearching rate, whereas the asymptotic ingestion rate is given by thereciprocal of the handling time. The reciprocal of the product of thesearching rate and the handling time is equivalent to the saturationcoefficient XK, which sets the prey density at which the ingestion rate ishalf the maximum rate.

87J. van der Meer / Journal of Sea Research 56 (2006) 85–102

value of {p˙Am} will depend on the diet and the ratio{p˙Am}/{JXm} gives the conversion efficiency of ingestedfood into assimilated energy.

The assimilated products enter a reserve pool and it isassumed that the reserve density [E] (energy reserves Eper unit of body volume V) follows first order dynamics,which means that the rate at which the reserve densitydecreases (in the absence of assimilation) is proportionalto the reserve density itself. Hence,

d½E�=dt ¼ �pA=V � c½E� ¼ �pAmf gfV�1=3 � c½E� ð1Þwhere c is the proportionality coefficient that sets therate at which the reserve density drops when noassimilation occurs. A maximum equilibrium reservedensity [Em] occurs at maximum food density, and canbe given (let d[E]/dt=0 and f=1) by [Em]={p˙Am}V

− 1/3/c. Hence the coefficient c in Eq. (1) depends on volume,and Eq. (1) can be re-written as:

d½E�=dt ¼ �pAmf gV�1=3 f � ½E�= Emf gð Þ ð2ÞIt follows that the equilibrium reserve density [E]⁎ isproportional to the scaled functional response: [E]⁎= f[Em]. The assumption of first order dynamics of reserve

density is one of the most important aspects of DEBtheory. It is also the most difficult part of DEB, since theproposed mechanism underlying this assumption israther complicated (Kooijman, 2000, pp. 246).

The utilisation rate p˙C (with dimension energy perunit time), which is the rate at which energy is utilisedfrom the reserves, can be written as the differencebetween the assimilation rate and the rate at which thereserves change p˙C=p˙A−dE/dt. According to the chainrule for differentiation, the rate of change of thereserves can be written as the rate of change of thestructural volume multiplied by the energy density plusthe rate of change of the reserve density multiplied bythe structural volume: p ˙C=p ˙A−dE/dt=p ˙A−d(V[E])/dt=p˙A− [E]dV/dt−Vd[E]/dt. Combined with Eq. (2),this gives

�pC ¼�pAmf g½E�V 2=3

Em½ � � E½ � dVdt

ð3Þ

When the energy density has reached equilibrium, thefirst term is exactly equivalent to the assimilation rate.The second term at the right-hand side is necessary toprevent dilution of the reserves due to growth.

A fixed proportion κ of utilised energy is spent ongrowth plus maintenance plus (for endotherms) heat-ing, the rest goes to development (for embryos andjuveniles) or to reproduction (for adults). Maintenancecosts p˙M=[p˙M]V are proportional to body volume(where [p˙M] are the maintenance costs per unit ofvolume), and heating costs p˙T={p˙T}V

2/3 are propor-tional to body surface area (where {p˙T} are the heatingcosts per unit of surface area). Thus

j �pC ¼ EG½ �dV=dt þ �pM½ �V þ �pTf gV 2=3 ð4Þ

where [EG] are the energetic growth costs per unit ofgrowth in structural body volume. Substituting Eq. (3)in Eq. (4) gives the growth equation

dVdt

¼ j �pAmf g½E�= Em½ � � �pTf gð ÞV 2=3 � �pM½ �Vj½E� þ EG½ � ð5Þ

Under constant food conditions the growth equationsimplifies to

dV

dt¼ jf �pAmf g � �pTf gð ÞV 2=3 � �pM½ �V

jf EM½ � þ EG½ � ð6Þ

This equation is mathematically equivalent to the VonBertalanffy growth model (Fig. 3). The ultimatevolumetric length, which is the cubic root of the

Fig. 3. Body length as a function of age. At constant food conditions,the DEB model predicts that volume growth follows the VonBertalanffy growth equation, where the growth rate is given by thedifference between a surface-area related term and a volume relatedterm. Using the chain rule dV/dt=(dV/dL)·(dL/dt)=3(δML)

2·dL/dtreveals the (non-autonomous first order) differential equation for theVon Bertalanffy length growth dL/dt=r˙B(L∞−L). This equation can besolved to obtain length as a function of time (or age), i.e. L(t)=L∞(1−exp(r˙Bt)).


ultimate volume, equals (κf{p˙Am}−{p˙T})/[p˙M], and theVon Bertalanffy growth coefficient equals

13

�pM½ �jf EM½ � þ EG½ �.

The Von Bertalanffy growth model is, however, basedon an entirely different biological rationale. DEBtheory uses the energy conservation law and assumesthat the difference between a supply (a fraction κ ofthe utilisation rate) and a demand term (the mainte-nance rate and the heating rate) is available forgrowth. Hence, the proximate control of growth is theutilisation rate, but the ultimate control is, of course,in resource intake by the individual from theenvironment (Fig. 1). In contrast, Von Bertalanffydid not apply the energy conservation law to theoverall organism, but defined growth as the differencebetween anabolism (synthesis) and catabolism(breakdown).

By combining growth Eqs. (5) and (3), the utilisationrate can now be written as an explicit function of bodyvolume and reserve density:

�p C¼ ½E�j½E� þ EG½ ��

�pAmf g EG½ �Em½ � þ �pTf g

� �V 2=3þ �pM½ �V

� �ð7Þ

Under constant food conditions (when the reservedensity quickly reaches an equilibrium and becomesproportional to the scaled functional response: [E]⁎= f[Em]), the utilisation rate is thus a function ofsurface-area and volume.

As was said above, a fixed proportion 1−κ of theutilised energy goes to development and reproduction.Embryos (which do not feed and do not reproduce)and juveniles (which feed but do not reproduce) usethe available energy for developing reproductiveorgans and regulation systems. DEB theory assumesthat the transitions from embryo to juvenile and fromjuvenile to adult occur at fixed sizes, Vb and Vp,respectively. The so-called κ-rule for allocation differsfrom other allocation rules in that no sudden change inthe somatic growth pattern occurs at maturity. The rulealso implies a similarity of growth patterns betweenthe sexes. DEB theory further assumes that thereproductive organs require so-called maturity main-tenance costs. Since the development stops at maturity(when the juvenile turns into an adult), these maturitymaintenance costs do not increase any further aftermaturation. DEB theory states that for embryos andjuveniles the ratio between the costs for developmentand the maturity maintenance costs is the same as theratio between the costs for somatic growth and the(somatic) maintenance costs. Hence the utilisation rateallocated to development (first term at the right-handside of Eq. (8)) and maturity maintenance (secondterm at the right-hand side of Eq. (8)) is for(ectothermic) embryos and juveniles given by:

1� jð Þ �pC ¼ 1� jj

EG½ �dV=dt þ 1� jj

�pM½ �V ð8Þ

The energy needed for maturity maintenance reaches itsmaximum at the size at maturity Vp. The size at maturity,i.e. the size at which a juvenile becomes an adult, isassumed to be constant. For adults, which no longer payfor development, the allocation to reproduction (firstterm at the right-hand side) and maturity maintenance(second term at the right-hand side) is given by:

1� jð Þ �pC ¼ �pR þ 1� jj

�pM½ �Vp ð9Þ

Combining Eqs. (7) and (9) will give an explicitexpression for the reproduction rate (of ectotherms) p˙R.At constant food conditions and when the animal hasreached maximum size, such expression simplifies to

�pR ¼ 1� jð Þf �pAmf gV 2=3l � 1� j

k�pM½ �Vp ð10Þ

Table 1The basic assumptions of the κ-rule DEB model

1. An organism is characterised by a structural body and a reserve density (i.e. amount of reserves per amount of structural body). The chemicalcomposition of both structural body and reserves is constant, which is called the assumption of strong homeostasis.

2. Each organism starts its life as an embryo (which does not feed and does not reproduce). When the embryo has reached a certain degree ofmaturation, it changes into a juvenile (which feeds, but does not reproduce). Similarly, a juvenile changes into an adult (which feeds andreproduces) when it exceeds a given threshold value.

3. Ingestion is proportional to the surface area of the organism and depends upon food density by a Holling type II functional response. Recall thatembryos do not feed.

4. A fixed fraction of the ingested food is assimilated and enters a storage pool, which is characterised by the reserve density.5. The regulation of the reserve density follows a first-order process.6. A fixed fraction κ of the utilisation rate goes to somatic maintenance, heating (for endotherms) and growth of the structural body (with a priority for

maintenance), and the rest goes to maturity maintenance and (for embryos and juveniles) maturity or (for adults) reproduction.7. Maintenance rate is proportional to structural volume and heating rate is proportional to the surface area of the organism.


An overview of the main DEB theory assumptions isprovided in Table 1.

2.2. Temperature

The DEB theory uses the Van't Hoff-Arrheniusequation to describe the dependency of physiologicalrates on temperature. This equation has its origin instatistical thermodynamics, where the behaviour of asystem containing a very large number of a single typeof molecules is predicted from statistical considerationsof the behaviour of individual molecules (Haynie,2001). In its basic form the Van 't Hoff-Arrheniusequation looks like

�k Tð Þ ¼ �klexp�Ea

kT

� �ð11Þ

where k ˙(T) is a reaction rate that depends upon theabsolute temperature T (in Kelvin), k ˙sub ∞ is a(theoretical) maximum reaction rate, which is thereaction rate when all molecules would react. Theterm exp(−Ea/(kT)) is the Boltzmann factor, which givesthe fraction of the molecules that obtain the criticalactivation energy Ea (in joules per molecule) to react.This fraction increases with increasing temperature. Theconstant k (not to be confused with the reaction rate k ˙) isthe Boltzmann constant and equals 1.38 10− 23 joule perdegree Kelvin. The Van 't Hoff-Arrhenius equation canalso be re-written in the form

�k Tð Þ ¼ �

k1expTAT1

� TAT

� �ð12Þ

where k ˙1 is the reaction rate at a reference temperatureT1, and TA the so-called Arrhenius temperature (whichequals Ea/k).

Glasstone et al. (1941) showed that the Van't Hoff-Arrhenius equation is approximate for bimolecular

reactions in the gas phase. Kooijman (1993, 2000)emphasises the enormous step from a single reactionbetween two types of particles in the gas phase tophysiological rates where many compounds are in-volved and gas kinetics do not apply. He thereforeregards the application of the Van't Hoff-Arrheniusrelation to physiological rates as an approximation only,for which the parameters have to be determinedempirically. For this reason, Kooijman prefers the useof an Arrhenius temperature instead of the use of anactivation energy, which would give a false impressionof mechanistic understanding. Similar to Kooijman,Clarke (2003, 2004) and Marquet et al. (2004) stress thatthe Van't Hoff-Arrhenius equation is just a statisticalgeneralisation, and they too conclude that at present westill lack a clear understanding of the relationshipbetween temperature and metabolism at the organismalscale.

2.3. Size and shape

The size of the structural body, quantified as volumeV, is one of the two basic state variables of the DEBmodel. All energetic processes are directly related to V.Maintenance rate, for example, is proportional tovolume V, whereas assimilation rate is proportional tothe so-called volumetric surface area V2/3. In practicestructural volume is not easily measured. Measurementsof length are usually much easier to obtain, and if anorganism does not change in shape during growth, theneach length measure can be used to predict volume byusing a calibration curve of the form

V ¼ dMLð Þ3 ð13Þ

where δM is a shape parameter, whose value, of course,very much depends upon the type of length measure Ltaken. For shells, for example, length may be measured


as height or width. Mammals may be measuredincluding or excluding the tail, etc.

2.4. Compound parameters and a dimensionlessrepresentation of the DEB model

The differential equations for energy density (Eq. (2))and structural volume (Eq. (5)) are the core of the DEBmodel. The dynamical behaviour of the system asdefined by these coupled equations (i.e., reservedynamics and growth) can be most easily understoodby using the dimensionless form of the equations. Usingthe dimensionless variables scaled energy densitye= [E]/[Em], scaled volumetric length l=V1/3/Vm

1/3

(where, for ectotherms, maximum volumetric lengthVm1/3 equals κ{p˙Am}/[p˙M]) and scaled time τ= t[p˙M]/[EG],

Eqs. (2) and (5) turn into:

deds

¼ gf � el

ð14Þ

and

dlds

¼ g3e� leþ g

ð15Þ

where the compound parameter g is given by the ratio[EG]/(κ[Em]). Similarly, the dimensionless form of theutilisation rate (expressed as the scaled energy flux orpower, i.e. �pC EG½ �

Em½ �L3m PM½ � ) is given byeg

eþ ggl2 þ l3� �

andthe dimensionless form of the assimilation rate by gfl2.

Apparently, the dimensionless compound parameterg is the only parameter that truly affects the dynamicsof the two equations system. This observation is inaccordance with the suggestion of Fujiwara et al.(2005) that information on g in the data comes fromthe autocorrelation in the size trajectory. The parameterg has been given the name ‘energy investment ratio’ asit stands for the energetic costs of new structuralvolume [EG] relative to the maximum available energyfor growth and maintenance κ[Em]. The parameter‘maximum energy density’ [Em], and the two com-pound parameters ‘maximum volumetric length’κ{p˙Am}/[p˙M] and ‘maintenance rate coefficient’ k ˙M=[p˙M]/[EG] scale the two state variables and time to theirdimensionless equivalents. The ‘maintenance ratecoefficient’ stands for the ratio of the costs ofmaintenance to structural volume synthesis. Onlyfour parameters are thus needed to fully characterisethe dynamics of the reserves and the structural body.This implies that observations on assimilation rate,reserve dynamics (including utilisation rate) and

growth alone do not suffice to enable estimation ofall five (for ectotherms) basic DEB parameters [EG],[Em], {p˙Am}, [p˙M], and κ (the list of basic parametersshould actually also include the parameters Vb and Vp,which indicate the fixed sizes at which the transitionsfrom embryo to juvenile and from juvenile to adultoccur, but usually these sizes can be observed directly).The scaled reproduction rate (at constant food densityand when the animal has reached its maximum size,see Eq. (10)) equals (1−κ)g(f 3−Vp/Vm), and theappearance of the parameter κ implies that informationon reproduction is also required for the estimation ofall five basic DEB parameters.

The ratio of the area-specific assimilation rate to themaximum energy density occurs regularly in DEBtheory (e.g. in Eqs. (3) and (5)) and has been called theenergy conductance ν˙={p˙Am}/[Em]. The inverse of thiscompound parameter ν ˙ can be interpreted as aresistance.

3. Alternative approaches

A dynamic description of the energy budget of anindividual organism is a logical follow-up of a staticdescription, which was the prevalent approach in the1970s and 1980s. Various alternative dynamic modelshave been proposed, such as the net-production models(Lika and Nisbet, 2000) and the metabolic theory ofecology (Brown et al., 2004b). Below I will brieflydiscuss the static descriptions and the various dynamicalternatives.

3.1. IBP studies

Energy budget studies received an enormous impetusfrom the International Biological Program (IBP) thatstarted in the 1960s. The flows of energy and matterinto, within and out of an individual organism weredivided into a number of separate fluxes. The mostimportant ones distinguished are ingestion (total uptakeof energy or mass), defecation (part of the ingestion thatis not absorbed, but leaves the gut as faeces),assimilation or absorption A (part of the ingestion thatcrosses the gut wall, i.e. the difference betweeningestion and defecation), growth dW/dt (part of theabsorption that is incorporated in the body tissue of theorganism), reproduction G (part of the absorption that isreleased as reproductive bodies), excretion E (part of theabsorption that is released out of the body in the form ofurine, or other exudates, with the exception ofreproductive bodies), and finally respiration R (part ofthe absorption that is released in association with the


oxidation of organic compounds, and thus causes a netloss of CO2). Assuming the conservation of energy andmass, the balance equation dW/dt=A− (G+E+R) hasplayed a key role in the IBP. Hitherto, many studies havefollowed this IBP recipe of constructing an energybudget for an individual animal. Since the budget mustbalance, a term particularly hard to measure was oftenfound by difference. In the Scope for Growth (SFG)approach, for example, all terms apart from growth andreproduction are measured (Bayne and Newell, 1983).The SFG, which is the difference between absorptionand excretion plus respiration, of a ‘standard’ animal(e.g. a blue mussel of 1 g dry mass) has been frequentlyused as an indicator of the ‘health’ of the ecosystem(Smaal and Widdows 1994; Widdows et al., 1995).

One of the major shortcomings of IBP-type energybudget studies is that the results are only descriptiveand very hard to generalise, even towards animals ofthe same species but different in size. Measuring theSFG of a single individual of a particular size will notelucidate the link between the energy budget and, forexample, the age-size relationship. Knowledge on therelations between all energy budget terms and bodymass of the individual is required. The terms that weredistinguished in the IBP approach have been chosenbecause they are relatively easy to measure, and notbecause their relationship to body mass could be easilyderived from first principles. Respiration, for example,reflects the costs of many different processes. Apartfrom the basal maintenance costs of the body, whichare, among other things, due to the maintenance ofconcentration gradients across membranes and theturnover of structural body proteins, it includes heatingcosts, costs that are directly coupled to the ingestion offood and costs coupled to the growth of body tissue.Whereas DEB assumes that all these costs depend in adifferent way on body size (e.g. maintenance costsrelate to body structural volume, whereas heating costsrelate to surface area), IBP-type studies usually applyallometric curve fitting to generalise the obtainedfindings to other size classes. A further complicatingfactor that is not accounted for in the IBP approach, isthat not all surplus energy available for growth willimmediately be used for growth of the structural part ofthe body. It may be stored temporarily in a reservetissue buffer. As various processes, such as ingestionand maintenance, will basically be related to thestructural part of the body, such distinction between ametabolically active structural part of the body andinactive reserve tissue seems to be a prerequisite for aproper understanding of energy budgets of individualorganisms.

3.2. Net-production models

Various alternative dynamic energy budget modelshave been constructed (Nisbet et al., 1996, 2004; Likaand Nisbet, 2000) that have been classified as net-production models. These models differ from Kooij-man's κ-rule model in that they first subtract themaintenance costs from the assimilated products, beforethey are allocated to other metabolic processes (growth,reproduction). Kooijman (2000, pp. 365) points tovarious theoretical problems with net-production mod-els, one of them being that non-feeding animals (e.g.embryos or animals experiencing starvation periods) dohave to pay maintenance costs anyway, which thusrequires that extra switches have to be built in to paymaintenance costs from the reserves under such condi-tions. One might argue that theoretical arguments do notsuffice, as models are always simplifications of the truth,and that the proof of the pudding is in the eating. Amajorchallenge is then to find out what sort of experimentsenable the selection of the most appropriate model(Noonburg et al., 1998).

3.3. Metabolic Theory of Ecology

The so-called metabolic theory of ecology (MTE)advocated by Brown and co-workers (West et al., 1997,2001; Gillooly et al., 2002; Brown et al., 2004b) hasreceived much attention over the last decade, despite alack of generality of the proposed mechanism and a lackof internal consistency in the description of the energybudget of the individual organism (Van der Meer, 2006).Basis of the theory is the idea that whole-organismmetabolic rate is limited by the internal delivery ofresources to cells. Resources have to be distributedthrough branching networks, and it was suggested thatthe fractal-like designs of these networks cause thesupply rate and hence the metabolic rate to scale as a 3/4power of body volume. Such a closed branchingnetwork is, however, at best applicable to a minorityof species. Apart for the vertebrates, closed networks donot exist in the animal kingdom. It was further assumedthat the metabolic rate not only equals the supply rate,but also the maintenance rate, defined as the powerneeded to sustain the organism in all its activities. At thesame time, the difference between supply and mainte-nance was assumed to be entirely used for theconstruction of new body tissue. Yet, you cannot haveyour cake and eat it, and, in fact, this ambiguity violatesthe first law of thermodynamics, as denoted byMakarieva et al. (2004). Another disadvantage of theMTE model is that it does not provide explicit


descriptions of some basic energy fluxes, e.g. the fluxtowards reproduction.

3.4. Species-specific models

All types of DEB models, whether it is the κ-ruleDEB model or the net-production models, arerelatively simple models aiming to have a wideapplicability, throughout the animal kingdom andpossibly beyond. This implies that species-specificaspects, such as, for example, the low feeding rate oflarval nematodes, are not covered by these models.Practically inclined biologists therefore tend to developtheir own species-specific models (Scholten andSmaal, 1998, 1999). A major problem with thisapproach is that such descriptions lack a commonbasis and usually contain many ad-hoc descriptions forthe various sub-processes. So, apart from theircomplexity, this lack of a common basis hampers acomparison between these models. I believe that ifspecies-specific aspects have to be taken into account,a ‘phylogeny’ of models is needed, where the κ-ruleDEB model could play the role of the commonancestor and each progeny contains its own peculiar-ities. A good example is the work of Jager et al. (2005)on nematodes. They observed that nematodes differfrom other animals in their initial growth being slowerthan expected on the basis of size alone, and were ableto explain this lower growth by low feeding rates ofthe larvae. The size of the mouth cavity appeared tolimit the feeding rate of the larval worms. Jager et al.(2005) incorporated this phenomenon into the κ-ruleDEB model by correcting the ingestion rate using asize-dependent stress factor.

4. Parameter estimation and the link with empiricaldata: a short review of experimental approaches

So far, various studies have tried to estimate the fivebasic DEB parameters (the surface-area specific assim-ilation rate {p˙Am}, the volume-specific maintenance rate[p˙M], the volume-specific costs of growth [EG], themaximum energy density [Em], and the fraction of theutilised energy spent on maintenance and growth κ), or,alternatively, the five compound parameters (the energyinvestment ratio g, the maintenance rate coefficient k ˙M,the energy conductance ν ˙, the maximum volumetriclength Lm, and the Von Bertalanffy growth coefficientr˙B) for one or a few specific (metazoan) species.Examples are studies of daphnids Daphnia magna andD. pulex (Evers and Kooijman, 1989), pond snailsLymnea stagnalis (Zonneveld and Kooijman, 1989),

blue mussels Mytilus edulis (Van Haren and Kooijman,1993), the flatfish species dab Limanda limanda, plaicePleuronectes platessa, sole Solea solea and flounderPlatichthys flesus (Van der Veer et al., 2001), thenematode species Caenorhabditis elegans, C. briggsaeand Acrobeloides nanus (Jager et al., 2005), and thedelta smelt Hypomesus transpacificus (Fujiwara et al.,2005). These six studies used both observational (field)data and experimental data.

In the experiments both the initial state of theorganism (age, size) and the state of the environment(food density, temperature) have been manipulated.Response variables include feeding rate, assimilationrate, respiration rate, growth rate and reproductiveoutput. Most DEB parameters, such as the fraction ofthe utilised energy spent on growth and maintenance,the maintenance rate per unit of volume, and themaximum energy density, cannot be measured directly.One problem is that conceptual model processes donot have a simple one-to-one relationship to themeasurable response variables. Respiration rate, asmeasured by oxygen consumption, for example, doesnot represent only maintenance costs, but also over-head costs of growth and reproduction. This impliesthat the estimation procedures are often quite complex.It appears that compound parameters, usually ratiosof the primary parameters, are often more easilyestimable.

Below, the experiments are categorised in sevenclasses, (a)–(g) (Table 2a, b). Each class will be shortlydescribed.

4.1. Size and the functional response

Using the relationship between the ingestion rate JXon the one hand and length L and food density X on theother hand

�JX ¼ �

JXmn o

dMLð Þ2 XXK þ X

þ e; ð16Þ

the surface-area-specific maximum ingestion rate {JXm}and the saturation coefficient XK can be estimated. Notethat the shape coefficient δMmust be known beforehand.The relationship simplifies when either size or fooddensity is kept constant. The assimilation efficiency isrequired to derive the assimilation rate from theingestion rate. This efficiency is assumed to be constantand can be estimated by means of the Conover-ratiomethod (Conover, 1966).

For bivalves, the ingestion rate is usually estimatedby determining the filtration rate and the foodconcentration in the filtrated water separately (Van

Table 2aAn overview of field data and experiments used for estimating DEB parameters

a b c d e f g

Longitudinal × × × ×Treatment None × (×) ×

initial size × × (×)food density × × × i ∅ ? ?temperature ×

Response intake rate ×respiration rate × × ×size/growth × × × × ×mass/growth × × ×cumulative reproduction ×composition ×survival time ×

(a) Treatment factors and response variables in various types of experiments (coded a–g). When food density is not a treatment factor, food levels caneither be kept at zero (indicated by the symbol∅) i.e. in case of a starvation experiment, be ad libitum (∞), be a known initial amount that is depletedin the course of the observations (i), or be unknown (?).


Haren and Kooijman, 1993). Assuming that thefiltration rate does not depend on food density, thesurface-area specific filtration rate can be estimated byusing

�JW ¼ �

JWn o

V 2=3 þ e; ð17Þ

where JW is the filtration rate, {JW} the surface-areaspecific filtration rate, and ε random observation error.Another feature of bivalve feeding is that it is oftenobserved that the saturation coefficient is not constant,but depends on the silt content of the water (Pouvreauet al., 2006). This issue can be taken into account bymore specific models of the feeding process, in whichit is assumed that the food-acquisition apparatus canbe temporarily clogged by silt particles (Kooijman,2006).

Table 2bAn overview of field data and experiments used for estimating DEB parame

Short description

a Size and/or food density on ingestion rate (or filtration rate; or assimb Food density and size on ingestion rate and respiration ratec1 Food density on growth curvec2 Food density on cumulative reproductiond Age of embryo within an egg on dry weight, reserve materials and re1 Changes in respiration rate, body size and body mass during a starv

e2 Length on body size changes during starvatione3 Initial size on survival time during starvationf Temperature on growth rateg Observational length and mass (wet weight) data

(b) The DEB parameters estimated using the experimental data from the ex(1989), F to Fujiwara et al. (2005), H to Van Haren and Kooijman (1993), JZonneveld and Kooijman (1989).

4.2. Size and/or food density versus oxygen consump-tion rate

Van Haren and Kooijman (1993) used the oxygenconsumption rate JO as an approximation of theutilisation rate. Using Eq. (7)

�pC¼ ½E�j½E� þ EG½ �

�pAm EG½ �Em½ � þ �pTf g

� �V 2=3 þ �pM½ �V

� �;

which for ectotherms experiencing constant foodconditions can be written as

�pC ¼ f Em½ � EG½ �jf Em½ � þ EG½ �

�pAmf gEm½ � V 2=3 þ

�pM½ �EG½ � V

� �ð18Þ

they arrived at JO ∝ (ν˙(δML)2 +km(δML)

3). Hence, byfitting the oxygen consumption rate as a linear function

ters

DEB parameters References

ilation rate) {JXm} or {JW} or {p ˙Am}, and XK E, H, V, Z[p˙M]/κ, ν/kM, g{JXm} HL∞, r˙B; kM, ν˙ and g F, H, Jκ, {p˙Am}, [p˙M] J

espiration rate ν˙, kM Zation period ν˙, kM, g/e0 H

[p˙M], [Em] Vν˙ or [p˙M] Zν˙, {p˙Am

}/[p ˙M] or ν˙, κ ZTA H, JδM, L∞ H, V, Z

periment types a–g, and references. E refers to Evers and Kooijmanto Jager et al. (2005), V to Van der Veer et al. (2001), and Z refers to

Fig. 4. Length growth rate as a function of body length for various foodconditions (f). The Von Bertalanffy growth coefficient r˙B is equivalentto the (negative) slope of the relation between the length growth rateand length dL/dt= r˙BL∞− r˙BL. The intercept r˙BL∞ is the initial lengthgrowth rate and is often indicated with symbol ω (Appeldoorn, 1982).DEB theory predicts that ultimate size is smaller at lower foodconditions, but that the Von Bertalanffy growth coefficient increaseswith decreasing food conditions.


(without a constant) of the volumetric surface area andthe volume, they obtain (by taking the ratio of the tworegression parameters) the ratio between the energyconductance ν ˙ and the maintenance rate coefficient km.Note that implicitly it is assumed that the part of the fluxfrom the reserves that is not respired (and that isincorporated in either somatic tissue or in gonads) isnegligible compared to the fraction that is actuallyrespired.

If the food conditions are not constant, things get a bitmore complicated. However, Van Haren and Kooijman(1993) used an experimental dataset in which ingestionrate instead of food density itself was measured. Theysubsequently used the relationship between utilisationrate on the one hand and ingestion rate and structuralsize on the other hand, which can be derived by insertingthe equation for the ingestion rate JX={JXm}fV

2/3 intoEq. (18). This reveals

�pC ¼�JX Em½ � EG½ ��

JXj Em½ � þ �JXm

� V 2=3 EG½ �

��pAmf gEm½ � V 2=3 þ

�pM½ �EG½ � V

� �ð19Þ

which can be re-written, using JO=ηp˙c (where η is aconversion coefficient that couples an oxygen flux to anenergy flux) and V1/3 =δM·L, as

�J O ¼ g

�pM½ �j

�JX�JX = dM dLð Þ2þg �JXm�

��m�

kMþ dM dL

� �ð20Þ

Hence, the terms η[p˙M]/κ, g{JXm} and the ratio of thecompound parameters ν ˙ and kM can be estimated.

It should be realised that if one is unwilling toassume that the part of the flux from the reserves that isnot respired (and that is incorporated in either somatictissue or in gonads) is negligible compared to thefraction that is actually respired, matters become morecomplicated.

4.3. Food density and growth and/or reproduction

If food conditions are more or less constant, growthcurves can be used to estimate ultimate volume, whichfor ectotherms equals κf{p ˙Am}/[p ˙M], and the VonBertalanffy growth coefficient, which equals13

�pM½ �jf EM½ � þ EG½ �. Note that both compound parameterscontain the scaled food density f. It is not always

appreciated that the Von Bertalanffy growth coefficient,often called the Von Bertalanffy growth rate, is not agrowth rate, but represents the (negative) slope of therelation between the length growth rate and the length.In fact, DEB theory predicts that within the same speciesthe Von Bertalanffy growth coefficient will increasewith decreasing food levels (Fig. 4).

If food conditions are variable over the lifetime of anorganism, it might be possible to estimate threeparameters, e.g. kM, ν˙ and g, from the size trajectory.The size trajectory can be obtained by numericalintegration of the two DEB differential equations(Fujiwara et al., 2005). The cumulative reproductionversus age can be obtained in the same way (both forconstant and variable food densities). Recall that atconstant food conditions and when the animal hasreached maximum size, the reproduction rate (ofectotherms) is given by Eq. (10) (Fig. 5).

4.4. Growth and respiration rates of an embryo

Zonneveld and Kooijman (1989, 1993) used data ongrowth and oxygen consumption rates of embryos ineggs. The idea is that embryos do not feed, but that theyuse their high initial reserves for growth and mainte-nance. This phenomenon considerably simplifies re-serve dynamics and reduces scatter related to variable

Fig. 5. Cumulative maturation and reproduction as a function of age.Scaled functional response is 1. At the age of maturity, i.e. when thestructural body volume reaches VP, the total investment in maturationhas reached its maximum. From that moment onwards reproductionstarts and the cumulative reproduction steadily approaches a diagonalasymptote, whose slope is given by Eq. (10) (see text).


food intake. For animals that do not feed (f=0) Eq. (2)simplifies to

d½E�dt

¼ ��pAmf gEm½ � E½ �V�1=3 ¼ � �m E½ �V�1=3 ð21Þ

This simplified version of the differential equation forenergy density and the differential equation for thestructural volume (i.e. Eq. (5)) can be (numerically)solved in order to predict the growth of the embryo, andthe decrease in reserve mass (which is proportional to[E]V) over time. Yolk mass was taken as equivalent tothe reserves (Zonneveld and Kooijman, 1993), but forthe pond snail reserves included glycogen, galactogenand proteins that are easily mobilised (Zonneveld andKooijman, 1989).

4.5. Starvation experiments

For starving animals the rate of change in energydensity is given by Eq. (21). If growth has also ceased,and structural volume V is constant, Eq. (21) can beeasily solved:

½E� ¼ ½E�0exp � �mV�1=3t �

ð22Þ

The utilisation rate for animals that do not feed and donot grow simplifies to p˙C=−Vd[E]/dt, and the oxygen

consumption rate (substituting Eqs. (21) and (22) andJO=ηp˙c) can then be written as

J�O ¼ g �m½E�0V 2=3exp � �mV�1=3t �

ð23Þ

This approach, which has been followed by Evers andKooijman (1989) and by Van Haren and Kooijman(1993) is not entirely consistent with the κ-rule, as it doesnot indicate for what purpose the difference between theenergy flux to growth andmaintenance (which accordingto the κ-rule should equal κp˙C=−κVd[E]/dt) and themaintenance requirements, which equal [p˙M]V, is used.

4.6. Temperature and growth

DEB theory assumes that for a particular species, allrates (e.g. ingestion rate, respiration rate, growth rate)can be described, within a species-specific tolerancerange, by an Arrhenius relationship using a singleArrhenius temperature. The rationale behind thisassumption is that if different processes were governedby a different Arrhenius temperature, animals wouldface an almost impossible task of coordinating thevarious processes. Van Haren and Kooijman (1993)used the shell length growth rates of larval mussels atdifferent food conditions and temperatures to estimatethe Arrhenius temperature.

4.7. Size, mass and shape

The relationship between some length measure (e.g.shell length in the case of bivalves) and wet mass can beused to estimate the shape coefficient δM. However,special attention should be paid to the role of reservesand gonads. The idea is that the mass or volume measureshould represent structural size, and should thus not beaffected by the reserve density or the gonads. Hence drymass or ash-free dry mass is certainly inappropriate. Wetmass can be used if reserves that have been used arereplaced by water. The same should hold for gonads. Ifnot, gonads may be dissected, but the physical removalof reserves will cause a problem, as they are partlystored in a variety of tissues. Specific density has to beestimated or known.

5. Estimation procedures and the blue mussel as anexample

In this section, part of the parameter estimationprocedure of Van Haren and Kooijman (1993) isrepeated to illustrate the application of two statistical


approaches (simultaneous regression by means ofweighted non-linear regression, and repeated measure-ments or time-series regression) not commonly appliedin ecology, but very helpful in estimating DEBparameters. Data were read from the published graphs,which may have caused some inaccuracies in theestimates. Van der Veer et al. (2006) provide a morecomplete set of DEB parameter estimates for variousbivalve species.

5.1. Simultaneous (non-linear) weighted least-squaresregression

For the blue mussel Mytilus edulis Van Haren andKooijman (1993) assumed a specific density of 1g cm−2

in order to derive structural volume V from wet massmeasurements and subsequently fitted the relationshipV=(δML)

3 +ε. They arrived at an estimate (±SE) of thecubic shape parameter δM

3 of 0.03692±7.59·10−5, butdid not indicate whether they used ordinary least squares(OLS) regression or weighted least squares (WLS)regression (Wetherill, 1986; see also Appendix B). Fromthe graph (Fig. 6) it is clear that the error varianceincreases with increasing length. An OLS regressionrevealed an estimate of the shape parameter of 0.03683±6.39·10−4, and a WLS (assuming that the error varianceis proportional to cubic length) revealed an estimate of0.03692±6.15·10−4. The difference between theseestimates is rather small.

Fig. 6. The relation between body wet mass and shell length for theblue mussel Mytilus edulis. The fitted curve is based on a weightedleast-squares procedure. The error variance clearly increases withlength and the weights were chosen proportional to shell length. Figureafter Van Haren and Kooijman (1993; Fig. 1).

Van Haren and Kooijman (1993) used experimentaldata from Winter (1973) to fit the function JW={JW}(δML)

2 +ε between filtration rate JW and shell length L.The parameter {JW} is an area-specific filtration rate.The shape parameter δM was assumed to be known.Since filtration rates also depend upon food concentra-tions, the estimated value for {JW} is only valid for theexperimental food concentration, which equalled 40·106

cells dm−3. Subsequently, they used other data fromSchulte (1975) andWinter (1973) to fit filtration rate as afunction of both food concentration and shell length.They used the function �

JW ¼ �JWm

� XK

XK þ XdMLð Þ2þe, which

follows from the assumption that the ingestion rate JX,which equals the filtration rate JW times the food densityX, follows Holling's type II functional response. Thefunction contains two estimable parameters, the maxi-mum area-specific filtration rate {JWm} and the satura-tion coefficient XK. The parameter estimates from thesecond experiment can be used to predict an area-specific filtration rate for the first experiment. Applying atemperature correction (the first experiment was per-formed at 12°C and the second experiment at 15°C),using an Arrhenius temperature TA of 7579K, revealedan estimate for {JW} of 0.499dm

3 h−1 cm−2, on the basisof the data from the first experiment. Using theparameters of the second experiment, the obtainedestimate for {JW} is (slightly) different and equals0.544dm3 h−1 cm−2.

However, if two or more functions contain commonparameters it is perfectly possible to apply a singleparameter estimation procedure. Suppose, for example,that two sets of data are available, and that for both sets adifferent (non-linear) equation has to be fitted, contain-ing one or more common parameters. The error varianceis not necessarily the same. Hence, a random variable Y isrelated to an independent variable X by a (non-linear)function f(X,b) and a random variable Z is related to X bya (non-linear) function g(X,b). The two functions containa single common parameter b whose value has to beestimated, and the two types of observations havedifferent variances, which are known up to a constant:varY=wYσ

2 and varZ=wZσ2. An estimate for the

parameter b can then be obtained by a Weighted Least-Squares estimation (Appendix B)), that is by minimising

SS bð Þ ¼Xi

Yi � f Xi; bð Þð Þ2wY

þXj

Zj � g Xj; b� �� 2wZ

:

In practice these constants wY and wZ are not known, buta commonly applied approach is to perform a two-stepprocedure. In the first step, the two equations are fittedseparately, and the estimated residual variances are used


for determining the constants wY and wZ. Theseconstants are subsequently used in the second step,which is the (simultaneous) Weighted Least-Squareestimation. Applying this procedure to the two above-mentioned data sets used by Van Haren and Kooijman(1993) resulted in an estimated maximum area-specificfiltration rate {JWm} of 0.525dm3 h− 1 cm− 2 (Table 3).

Similarly, the data from Kruger (1960) on oxygenconsumption rate versus shell length (Van Haren andKooijman, 1993, their Fig. 8) and from Bayne et al. (1987,1989) on oxygen consumption rate versus ingestion rateand shell length (VanHaren andKooijman, 1993, their Fig.9) could have been used simultaneously (Table 3, Fig. 7).

5.2. Longitudinal studies, repeated-measurements ana-lysis and time-series regression

Quite often eco-physiological experiments are so-called longitudinal studies, which means that the responseis not a single observation in time, but consists of multiple(or even continuous) observations in time. For example,respiration rate might be repeatedly measured over aprolonged period of growth or starvation. Similarly,treatment conditions may vary over time. Food density,for example, is usually kept at a constant level, but

Table 3Estimates of DEB parameters for the blue mussel, based on Van Haren and

Data Original esti

1 Length and wet weight data δM3 =0.03692

2 Size on filtration rate {JW}=0.041dm3 h−1 cm−

3 Food density and size on filtration rate {JWm}=0.83dm3 h−1 cm−

XK=76 106±cells dm−3

2&3combined

4 Size on oxygen consumption rate ν˙/kM=26.5±5 Size and ingestion rate on oxygen consumption

rateη[p˙M]/κ=0.0cm3 O2 h

−1

cν˙/kM=5.3±6g{JXm}=0.1cm−2

h−1

4&5combined

planned fluctuating food levels have been used as well.Field data often consist of longitudinal data, e.g. observedweight loss during periods of natural starvation.

Repeated-measurements studies, such as observa-tions on the growth of individual organisms, have beenanalysed in two fundamentally different ways in theliterature (Sandland and McGilchrist, 1979). The‘statistical’ approach treats the repeated measurementson each individual as a multivariate observation orprofile (Johnson andWichern 1988) that can be analysedby a multivariate analysis of variance (MANOVA).Quite often, a further assumption is made on thedependence structure of the observations (i.e. so-calledcompound symmetry of the error covariance matrix isassumed, which means that all covariances are equal foreach pair of years), which allows for the use of aunivariate repeated measures analysis of variance(Winer, 1971; Potvin and Lechowicz, 1990). This‘statistical’ approach thus allows for the dependencestructure, but will leave a biologist, who requires aninterpretation of the analysis of variance coefficients,unsatisfied. Alternatively, the ‘biological’ approach fits abiologically meaningful model, such as the DEB growthmodel, through the individual observations, and differentparameters can be estimated for the different treatment

Kooijman (1993)

mates My estimates

±7.59·10−5 δM3 =0.03692±6.15·10−4 (using WLS with lengthas weights)

±0.0006752

{JW}=0.378±0.00522dm3 h−

1

cm−2

After temperature correction to 15 °C:{JW}=0.499dm

3 h−1

cm−2

±0.0982

{JWm}=0.831±0.101dm3 h−

1

cm−2

42 106 XK=73.6 106±41.7 106 cells dm−3

{JWm}=0.831±0.0837dm3 h−

1

cm−2

XK=68.5 106±27.2 106 cells dm−3

This gives: {JW}=0.525dm3 h−

1

cm−2

14.8mm ν˙/kM=7.67±15.52mm56±0.025m−3

η[p˙M]/κ=0.072±0.041cm3 O2 h

−1

cm−3

.5mm ν˙/kM=0.27±0.54mm6±0.07mg POM g{JXm}=0.22±0.13mg POM cm−2

h−1

Equal weighing of 4 and 5: η[p˙M]/κ=0.0098±0.0148cm3 O2 h

−1

cm−3

ν˙/kM=6.69±10.71mmg{JXm}=0.056±0.159mg POM cm−2

h−1

Non-equal weighing of 4 and 5:η[p˙M]/κ=0.0511±0.0201cm

3 O2 h−1

cm−3

ν˙/kM=0.672±0.543mmg{JXm}=0.161±0.0664mg POM cm−2

h−1

Fig. 7. Oxygen consumption rate as a function of (a) shell length and(b) ingestion rate and shell length for the blue musselMytilus edulis. Inpanel (b) squares indicate a shell length of 4.5cm, circles a shell lengthof 2.5cm. Curves are given by Eq. (20). Solid lines refer to theparameter estimates when the two datasets were used separately.Dotted and dashed lines refer to the jointly estimated parameters, usingdifferent weights. See also Table 3. Figure after Van Haren andKooijman, (1993; Figs. 8 and 9).

Fig. 8. Shell length of the blue mussel Mytilus edulis in theOosterschelde estuary during 1985 and 1986. Figure after Van Harenand Kooijman (1993, Figs. 14 and 15).


levels. The data are related through time and thedependence structure in the underlying (growth) processcan be explicitly taken into account by introducing aprocess-error term, or it can be ignored by treating allerror as observation error (Priestley, 1981; Harvey,1993; Hilborn and Mangel, 1997). If there is random-ness in the underlying (growth) process, then error willpropagate through time. Faster growth than expectedduring a certain time period will have its effect on length(and perhaps on growth) at later stages. Observation

error, on the other hand, will not have any effects onlength later on. The animal does not know what sort ofobservation errors we make. Excluding one type ofnoise in the analysis might lead to biases in parameterestimation (Hilborn, 1979; Quinn and Deriso, 1999).Recently, the so-called numerically integrated state-space method (NISS) has been advocated as being ableto incorporate both types of error simultaneously(Kitigawa, 1987; De Valpine and Hastings, 2002).This method uses two models, one for the processincluding process error, the other for the observations,including observation error. Fujiwara et al. (2005) usedthis method to analyse the growth of the delta smeltunder variable food conditions. They introduced processnoise in terms of a, for each individual independently,randomly fluctuating food density, governed by astochastic process with two unknown parameters.Here, Van Haren and Kooijman (1993) is followedand it is assumed that all error is observation error,which considerably simplifies analysis. For each set ofparameter values the two differential equations of theDEB model are numerically integrated, and the modelpredictions concerning the size trajectory are comparedto the observations by means of the residual sum ofsquares. A non-linear least-squares optimisation proce-dure (the Gauss-Newton method) is used to find that setof parameter values that minimise the sum of squares. Itis impossible to estimate all DEB parameters on thebasis of the size trajectory alone and even when most ofthe parameters were assumed to be known from othersources, the data were not appropriately fitted (Fig. 8).


Finally, it should be noted that in some cases,longitudinal studies are not repeated-measurementsstudies in statistical terms. If individual organisms aretreated separately and measured only once, but atdifferent points in time, then the observations areindependent. Similarly, if only two observations aremade in time (an initial observation and a finalobservation), then the difference between these tworepeated measurements might be taken as the responsevariable (e.g. a growth measurement, instead of tworepeated size measurements). This way, dependencieswithin an individual organism are implicitly taken intoaccount.

6. Discussion

The six papers reviewed (Table 2b) showed a varietyof approaches for estimating the DEB parameters. Noneof the papers has been able to obtain reliable estimatesfor all five basic parameters or, alternatively, for the set ofcompound parameters. Apparently, DEB parameters arenot easy to estimate. A reason for this problem is that oneof the two state variables, reserve density, is extremelyhard to measure. The only reviewed paper in whichreserve density was measured directly is Zonneveld andKooijman (1989), but they considered the special case ofembryo eggs, where the reserves are relatively easy tomeasure. Van der Veer et al. (2001) indirectly measuredthe maximum reserve density by assuming that flatfishreached their maximum reserve capacity at the start of anatural starvation period and lost all their reserves at theend of the period. An alternative approach wasadvocated by Van der Meer and Piersma (1994), whoapplied the idea of strong homeostasis to distinguishbetween reserves (what they called stores) and structuralbody. Strong homeostasis means that the chemicalcomposition (for example, in terms of fat mass versusnon-fat mass) of both structural body and reserves isconstant, but not necessarily the same. For several birdspecies they analysed carcasses of a large number ofindividuals, including severely starved ones, in terms offat versus non-fat mass. They observed that (aftercorrecting for size) the relationship between lean massand fat mass could be described by a two-piece (orbroken) linear regression. The slope of one piece of theregression line represents the composition of thereserves, whereas the slope of the other piece revealsthe composition of the structural body. Animals aroundthat second piece had already started to break down theirstructural body. The breakpoint between the two piecesrepresents the structural body. Knowing the relationbetween size and structural bodymass enables prediction

of the actual reserves by subtracting the predictedstructural body mass from the observed mass. Theapproach fails when reserves are replaced by water, asoccurs in many aquatic animals.

If indeed observations on reserve density are lacking,it will be impossible to estimate all DEB parameters onthe basis of information on food input, temperature andthe size trajectory alone. Additional information on thevarious energy fluxes (such as assimilation, respirationand reproduction) is needed as well. From the re-analysisof the blue mussel data (Van Haren and Kooijman, 1993)it appears that severe estimation problems may occurwhen such data on energy fluxes are analysed inisolation. Even using the available data sets on oxygenconsumption rate versus shell length and ingestion ratesimultaneously, resulted in extremely large standarderrors (Table 3, Fig. 7). Using a slightly differentprocedure of weighing the two data sets revealedcompletely different parameter estimates. Using asmuch information as possible in a single estimationprocedure can reduce this problem of overfitting.

Writing the two DEB differential equations in adimensionless form already yields some hints for a ruleof thumb of how to estimate the basic DEB parameters.The dimensionless form only contained the parameter g,which is the energy investment ratio. Hence, thisparameter might be estimated from the wiggles in thesize trajectory, particularly when the scaled functionalresponse varies over time in a known way (Fujiwara etal., 2005). Three other parameters, viz. maximumenergy density and the two compound parametersmaximum volumetric length and maintenance ratecoefficient, are needed to scale the two state variablesV and [E] and the variable time t to their dimensionlessequivalents. All five basic DEB parameters can bederived from estimates of these three parameters and ofthe parameters g and κ. The maximum energy density[Em] could be determined by providing animals with adlibitum food for a period long enough for the reservedensity to reach the maximum reserve density. Subse-quently, animals are starved for variable periods of timeand the change in composition (e.g. in terms of fat, drylean mass, and water) over time allows the estimation ofboth the structural body size and the maximum energydensity (Van der Meer and Piersma, 1994). Maximumvolumetric length Vm

1/3 and the maintenance ratecoefficient kM follow from the size trajectory of animalsthat have been able to grow up under ad libitum foodconditions (if g is known, as the reciprocal of the VonBertalanffy growth coefficient under ad libitum foodconditions equals 3/(gkM)+3/kM). The parameter κfollows from the reproduction rate of animals that


have reached (at constant and known scaled functionalresponse) their maximum size, see Eq. (10). Hence, inprinciple no information on assimilation rates andrespiration rates is needed to arrive at estimates of thebasic DEB parameters. Although the measurements ofthese rates can be problematic (Van der Meer et al.,2005) and although the use of respiration rate as anapproximation of the utilisation rate is not entirelycorrect, additional information on these rates may leadto more accurate parameter estimates. Hence, apart fromknowledge on (long-term) growth trajectories andcumulative reproduction obtained under controlled (orat least known) conditions, an additional short-termexperiment, in which both food conditions (using bothconstant and time-varying food conditions, rangingfrom zero to ad libitum) and temperature are varied in asystematic way, and in which assimilation, respiration,reproduction, size and body composition are measuredat regular intervals, would contribute much in revealingreliable estimates of the basic DEB parameters.

Knowledge of the basic DEB (compound) para-meters over a wide range of species opens opportunitiestowards a more quantitative understanding of the broadpatterns in physiological diversity, for example in termsof phylogenetic relatedness and ecology (Spicer andGaston, 1999).

Appendix A. A list of important DEB parameters

DEB uses a notation that is helpful in a quickinterpretation of the equations. All rates have dots,which indicate that they contain the dimension ‘pertime’. All variables that are expressed per unit volumeare given between square brackets. All variables that areexpressed per unit surface area are given betweenbraces. Notation from Kooijman (2000) is used. Theearlier papers and books (Kooijman, 1986a,b, 1993)used slightly different notations, shown in the columnPrevious symbols.

Symbol
Dimension Interpretation Previoussymbols
[E]
eL−3 Energy density L L Body length V L3 Structural body volume F – Scaled functional response T T Temperature X # l − 2 o r
#l−3
Food density in theenvironment
[EG]
eL−3 Volume-specific costs of growth η, [G] [Em] eL−3 Maximum energy density [ S m ] ,
[Em]
{JXm} #L−
2

t−1

Surface-area-specificmaximum ingestion rate

⌊I·m⌋,{I·m}

{p˙Am}
eL−2
t−1

Surface-area-specificmaximum assimilation rate

⌊Am⌋,{Am}

[p˙M]
eL−2
t−1

Volume-specific maintenance rate
ς˙, [M˙ ] {p˙T} eL−
2

t−1

Surface-area-specific heating rate
TA T Arrhenius temperature XK #l−
2

or #l−3

Saturation coefficient
K δM – Shape (morph) coefficient κ – Fraction of utilisation rate spent on
maintenance plus growthE½ �

g
– Energy investment ratio G
j Em½ �
k˙ M t−
1

Maintenance rate coefficient�pM½ �EG�
m˙
ν˙
Lt−1
Energy conductancepAmf gEM

Lm
L Maximum volumetric lengthj �pAmf g � �pTf g�pM½ �
L∞
L Ultimate volumetric lengthjf �pAmf g � �pTf g�pM½ �
r˙B
t−1
Von Bertalanffy growth coefficient13

�pM½ �jf EM½ � þ EG½ �

Appendix B. Weighted non-linear regression

B.1. Ordinary Least-Squares Regression

In Ordinary Least-Squares regression (OLS) theparameters of a linear regression model are estimatedby minimising the sum of squares. In case of, forexample, the regression-through-the-origin modelYi=bXi+εi, the problem is to find b that minimises thesum-of-squares function SSðbÞ ¼ P

i Yi � bXið Þ2.The solution to this problem is obtained by setting the

derivative of the sum-of-squares function to b equal tozero. This reveals the so-called normal equation:

ASSðbÞAb

¼ �2Xi

Xi Yi � bXi

� � ¼ Xi

XiYi � bXi

X 2i

¼ 0 ; resulting in b ¼P

XYPX 2

:

The minimum sum of squares divided by n−m, wheren is the number of observations and m the number ofparameters (i.e. m=1 in the present case), provides anunbiased estimate σ3 of the variance of the observa-tions. The variance of a linear function of independentrandom variables is given by var(uY1+vY2)=u

2 varY1+v

2 var Y2. Using this equality, it follows that

var b ¼P

X 2varYPX 2ð Þ2 ¼ r2P

X 2:


B.2. Weighted Least-Squares Regression

The rationale for using Weighted Least-Squaresregression (WLS) is usually the fact that the observa-tions do have different variances, which are known up toa constant: varYi=wiσ

2. The parameters of, for example,the regression-through-the-origin equation Yi=bXi canthen be obtained by minimising

SS bð Þ ¼Xi

Yi � bXið Þ2wi

:

For convenience, the problem can be re-stated in termsof an OLS problem, by dividing both sides of theregression equation by

ffiffiffiffiffiwi

p. This gives a regression

model with YiV¼ Yi=ffiffiffiffiffiwi

pas the dependent variable, and

XiV¼ Xi=ffiffiffiffiffiwi

pas the independent variable. The OLS

procedure then minimises

SS bð Þ ¼Xi

YiV� bXiVð Þ2¼Xi

Yiffiffiffiffiffiwi

p � bXiffiffiffiffiffiwi

p� �2

¼Xi

Yi � bXið Þ2wi

;

and is thus equivalent to WLS.

B.3. Non-linear Least-Squares Regression

The problem with non-linear equations is that thenormal equations cannot be solved analytically. Forexample, in case of the model Yi=aXi

b+εi, the normalequations are given by

ASSða; bÞAa

¼ �2Xi

X b i Yi � aX b

i

�¼ 0

ASSða; bÞAb

¼ �2Xi

log Xið ÞaX b i Yi � aX b

i

�¼ 0;

which illustrates their intractable nature, i.e. the equa-tions cannot easily be solved for â and b. Numericaliterative methods are required to find the least-squaressolution. See, for example, Seber and Wild (1989) foran introduction to non-linear regression.

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Documents

An introduction to Dynamic Energy Budget (DEB) …dimension unit energy per unit time) can thus be written as p˙A ={p˙ Am}fV2/3,where{p˙Am }isthemaximum surface-area-specific assimilation